\ 


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PERSPECTIVE 


A 


Series  of  Elementary  Lectures 


BY 

ADA  CONE, 


Late  Instructor  in  Massachusetts  State  Art  Schools  ;  Super¬ 
visor  OF  Drawing  in  Concord,  N.  H.,  Public  Schools, 

AND  Lecturer  on  Industrial  Art  to  Teachers’ 
Institutes  ;  Pupil  Museum  of  Fine 
Arts,  Boston. 


NEW  YORK : 

WILLIAM  T.  COMSTOCK, 

23  Warren  Street. 

1889 


COPYRIGHT, 

A.  L.  CONE, 

1889. 


PREFACE. 


rriHOUSA^^DS  of  people  are  giving  attention  to 
art  nowadays,  who  would  be  glad  of  some 
knowledge  of  perspective  if  they  could  have  it  pre¬ 
sented  with  sufficient  clearness  and  simplicity  to  make 
it  come  within  easy  comprehension,  and  the  work  of 
numberless  artisans  suffers  for  lack  of  a  concise  and 
simple  manual  on  this  subject.  These  lessons  are  ad¬ 
dressed  especially  to  these  and  other  adults  who 
have  either  no  time  or  no  inclination  to  go  into  the 
details  of  the  science  or  to  follow  it  to  a  far  analysis. 
I  have  developed  with  a  care  which  I  have  not  seen 
heretofore  in  any  elementary  work  those  points 
which  experience  tells  me  are  the  greatest  stumbling 
block  to  beginners,  and  some  of  the  illustrations 
are  novel. 


THE  AUTHOR. 


INTRODUCTION. 


Perspective  lias  been  considered  one  of  the  most 
difficult  subjects  which  come  within  the  special  range 
of  the  art  student.  The  more  books  have  accumu¬ 
lated  purporting  to  explain  its  elements  the  more  com¬ 
plicated  it  has  become,  until  the  luckless  pupil  is 
involved  in  a  labyrinth  of  lines  whose  directions  he 
does  not  understand  and  whose  endings  are  goals  at 
which  his  mind  never  arrives.  “  What  awful  perspec¬ 
tive!”  exclaimed  Wordsworth  as  he  looked  down  the 
aisles  of  King’s  College  Chapel,  and  ‘‘What  awful 
perspective  !  ”  echoes  the  teacher  by  methods  in  vogue 
as  he  walks  down  the  school-room  aisles  on  a  perspec¬ 
tive  recitation  day  and  observes  retreating  lines  pro¬ 
duced  impartially  to  M.  Ps,  S.  Ps,  and  other  Ps  not 
laid  down  on  the  ordinary  diagram,  a  drawing  here 
and  there  recalling  the  line  of  a  late  newspaper  poet : 

“And  see  where  lines  diverging  meet.” 

The  best  treatises  on  the  subject  are  too  abstruse  for 
elementary  pupils,  and  those  which  have  attempted 
simplification  are,  for  the  most  part,  too  vague.  Ele¬ 
mentary  works  have  consisted  of  sets  of  drawings  with 
explanatory  text  rather  than  of  analyses  of  principles 
with  diagrams  by  way  of  illustration,  and  have  had 
for  this  reason  little  educational  value,  the  mind  instead 
of  being  concentrated  upon  important  truths  being 
distracted  by  particular  problems  without  general  appli- 


8 


PEE8PECTIVE. 


cation.  As  a  result  many  persons  whose  work  calls  for 
practical  application  of  this  science  remain  ignorant  of 
its  primary  principles. 

Perspective  refers  to  the  appearance  of  objects  as 
influenced  by  position  and  distance  from  the  eye. 
Facts  which  are  local  and  ascertained  by  actual  meas¬ 
urement,  make  when  delineated  what  are  called  geo¬ 
metric  or  working  drawings.  Such  are  used  by  the 
artisan  for  patterns.  Perspective  drawings  give  the 
appearance  from  one  point  of  view,  as  is  necessary  in 
pictures.  If  we  see  three  sides  of  a  cube  and  draw  the 
geometric  facts  of  what  we  see,  the  drawing  will  look 
like  Fig.  1,  but  it  does  not  appear  thus ;  it  appears. 


Fig.  1 

instead,  like  Fig.  2.  By  this  we  know  that  there  is  a 
science  of  appearances ;  and  the  fact  that  every  object 
which  occupies  space,  be  it  a  house  or  a  flower,  an 
avenue  of  trees,  or  a  row  of  preserves  on  a  shelf,  in¬ 
volves  the  aid  of  this  science  to  represent  makes  knowl¬ 
edge  of  it  imperative  to  the  draughtsman. 


PERSPECTIVE. 


9 


The  following  lessons  state  simply  the  important 
truths  and  make  several  practical  applications,  going 
very  little  into  the  mathematical  part  of  the  subject, 
but  giving  information  in  such  a  form  as  to  be  of  easy 
application  to  the  artisan  and  the  amateur. 


CHAPTEE  I. 

Pkin.  1. — P amllel  retreating  lines  converge. 

For  proof  of  this  stand  in  the  street  and  observe  the 
rows  of  buildings  on  either  side ;  as  they  retreat  they 
tend  toward  each  other :  or  stand  on  a  railroad  track 
and  observe  the  same  phenomenon ;  the  rails  as  they 
retreat  converge.  Place  yourself  under  an  elevated 
railroad  and  observe  the  lines  above  and  the  lines  of 
the  surface  track  below,  the  lines  on  cither  side  of 
pillars  and  spandrels,  all  converging  towards  a  common 

point  in  the  distance.  An 
open  door  is  a  convenient 
illustration;  the  top  and  bot¬ 
tom  do  not  appear  parallel,  as 
they  really  are,  but  conver¬ 
gent. 

Parallel  lines  which  do  not 
retreat  do  not  appear  to  con¬ 
verge  but  to  remain  parallel. 
Take  for  example  the  nearest 
side  of  the  cube.  Fig.  2.  It 
is  a  square  and  has  parallel 
lines — two  vertical  and  two 
horizontal — but  these  lines  do 
not  retreat  and  therefore  re- 
Fis.3  main  parallel  in  appearance. 

As  soon,  however,  as  they 


PERSPECTIVE. 


11 


were  placed  in  a  retreating  position,  they  would  appear 
to  converge. 

This  convergence  causes  the  farther  side  of  an  object 
to  appear  smaller  than  the  nearer  side.  For  if  the 
retreating  lines  of  an  object,  as  the  top  and  bottom  of 
a  door,  being  parallel,  appear  to  converge,  then  the 
farther  upright  line,  being  of  the  same  length  as  the 
nearer  one,  must  appear  shorter,  in  order  to  meet  the 
ends  of  the  apparently  converging  lines. 

If  an  object,  as  a  stick,  is  moved  into  the  distance 
its  ends  will  describe  two  parallel  retreating  lines. 


which  lines,  according  to  the  law  of  perspective, 
appear  to  converge.  In  other  words  the  stick  grows 
smaller  constantly  as  the  distance  increases.  (See 
Fig.  4.)  Of  a  row  of  houses  equal  in  size,  the  house 
nearest  the  spectator  will  appear  largest,  and  they  will 
decrease  in  size  as  the  distance  increases  (Fig.  5). 

The  statement  of  the  principle  may  be  varied  thus  : 

The  apparent  size  of  objects  is  according  to  the 
square  of  the  distance. 

There  is  a  common  point  where  parallel  retreating 
lines  meet  if  produced.  Continue  for  illustration  the 


12 


PEESPECTIVE. 


retreating  lines  in  any  one  of  the  drawings  given  and 
it  will  be  seen  that  they  have  a  common  centre  ;  note 
that  the  rails  on  the  track  come  together  in  the  dis¬ 
tance  ;  observe  also  that  the  lines  seem  to  rise  as  they 
retreat ;  that  the  lines  of  the  elevated  track  above  seem 
to  tend  downwards ;  that  the  left  side  of  the  street 


tends  towards  the  right,  the  right  towards  the  left, 
and  all  towards  a  point  directly  opposite  the  eye. 

Ketreating  lines  whether  above  or  below  the  eye, 
tend  towards  the  level  of  the  eye.  Parallel  retreating 
lines  meet  at  the  level  of  the  eye.  The  point  where 
parallel  retreating  lines  meet  is  called  the  Yanishing 
Point. 


PEESPECTIVE. 


13 


Lines  which  run  directly  back  from  the  spectator, 
as  do  all  the  retreating  lines  in  the  drawings  to  this 
lecture,  retreat  at  an  angle  of  90°.  When  an  object 
is  so  placed  that  its  retreating  lines  are  at  this  angle 
the  object  is  said  to  be  in  parallel  perspective. 

Lines  which  retreat  at  an  angle  of  90°  find  their 
Vanishing  Point  directly  opposite  the  eye. 

To  make  a  perspective  diagram,  draw  first  a  hori¬ 
zontal  line  of  indefinite  length.  This  is  called  the 
Horizontal  Line  and  represents  the  level  of  the  eye. 
Objects  which  are  above  or  below  the  level  of  the  eye 
are  drawn  in  a  corresponding  position  with  regard  to 
this  line.  Make  a  dot  at,  or  near,  the  centre  of  the 
line.  This  represents  the  point  directly  opposite  the 
eye,  and  is  called  the  Centre  of  Vision  (C.  Y.),  or, 
sometimes,  Point  of  Sight  (P.  S.)  It  is  the  Vanishing 
Point  for  all  lines  which  retreat  at  an  angle  of  90° 
and  should  therefore  be  marked,  also,  Y.  P.  Draw  a 
line  from  this  point  at  right  angles  to  the  Horizontal 
Line,  indefinite  in  length.  This  represents  the  prin¬ 
cipal  visual  ray,  called  also  the  Line  of  Direction. 
The  end  of  this  line  is  the  position  of  the  eye  of  the 
spectator,  and  is  called  the  Station  Point  (S.  P.) 

Obtain  two  other  points  in  the  following  manner : 
Measure  the  distance  from  the  Y.  P.  to  the  S.  P.  and 
mark  points  the  same  distance  from  the  Y.  P.  each 
side  on  the  Horizontal  Line.  These  are  called  Meas¬ 
uring  Points  (M.  Ps).  They  may  be  most  easily 
located  by  means  of  compasses,  as  seen  at  Fig.  12, 
thus :  Take  the  Y.  P.  as  a  centre,  and  the  distance  to 
the  S.  P.  as  a  radius,  and  draw  an  arc  to  cut  the  Hor¬ 
izontal  Line  on  each  side ;  the  intersections  will  be  the 
M.  Ps. 


14 


PERSPECTIVE. 


Having  the  diagram  made  proceed  to  draw  the  ob¬ 
ject.  Begin  always  by  drawing  its  nearest  side,  or 
edge,  the  part  which  is  unchanged  by  perspective  and 
which  requires  only  to  be  placed  on  the  diagram  in  its 
requisite  geometric  relation  to  the  lines  of  the  diagram. 
Draw  next  the  retreating  sides  towards  their  Y.  P. 
Fig.  6  shows  the  nearest  side  of  a  cube.  The  cube  in 
this  position  is  below  the  level  of  the  eye  because  it  is 
below  the  Horizontal  Line  which  represents  the  level 


S.P. 


of  the  eye  on  the  diagram.  It  is  also  to  left  of  the 
spectator  because  it  is  to  left  of  the  Principal  Yisual 
Bay.  Being  in  this  position  we  are  able  to  see  the 
top  and  one  upright  retreating  side.  These  retreat  at 
an  angle  of  90°,  their  direction  is,  therefore,  towards 
the  C.  V.,  which  is  their  Vanishing  Point. 

How  that  the  retreating  lines  are  drawn,  w^e  come 
to  Prin.  2,  for  the  question  arises, — where  shall  we 
place  the  farther  vertical  and  horizontal  lines  to  com¬ 
plete  the  object  ?  For  we  have  seen  by  Figs.  1  and  2 


PERSPECTIVE. 


15 


that  it  will  not  do  to  draw  these  lines  as  far  off  as  they 
really  are. 

Hold  a  sheet  of  paper  directly  facing  you  and  it 
looks  as  wide  as  it  is,  turn  it  a  little  from  you  and  it 
appears  narrower,  and  if  ‘it  is  turned  as  far  round  as 
is  possible  it  is  narrowed  to  a  straight  line.  From 
this  we  get 


Prin.  2. — Objects  seen  obliquely  are  foreshortened. 

The  degree  to  which  the  retreating  sides  are  fore¬ 
shortened  depends  upon  the  position  of  the  object. 
The  nearer  the  object  approaches  the  Principal  Visual 


M.P.l 


V.P. 

C.Y. 


H.L.  M.P.2 


G.S. 


Fig.  T 


S.P. 


Pay  the  narrower  the  vertical  retreating  side  becomes, 
until  on  this  line  it  is  a  single  line  and  top  and  hot. 
tom  retreating  lines  are  one.  The  same  phenomenon 
occurs  as  the  horizontal  retreating  side  approaches  the 
level  of  the  horizontal  line. 

Perspective  measures  the  foreshortening  in  this 
way  ; 

Produce,  as  in  Fig.  7,  the  ground  line  A  from  the 
nearest  ground  angle  B  indefinitely.  Measure  from 


16 


PEKSPECTIVE. 


the  angle  B  out  on  the  produced  ground  line  a  dis¬ 
tance  equal  to  the  actual  width  of  the  adjacent  side, 
whose  perspective  width  it  is  desired  to  find.  This 
measurement  is  called  a  Geometric  Scale  (G.  S.)  From 
the  end  of  this  scale,  C,  draw  a  line  to  the  M.  P., which 
is  on  the  same  side  of  the  diagram  as  the  cube.  The 
place  where  this  line  crosses  the  retreating  ground 
line,  D,  is  the  farther  ground  angle,  E.  This  deter- 


v.p. 


G.S. 

Fig.  8 


SP. 

mines  the  perspective  width  of  the  side  in  the  posi¬ 
tion  in  which  it  is  placed.  Baise  the  vertical  line  to 
meet  the  upper  retreating  line.  This  intersection,  H, 
gives  one  of  the  farther  angles  for  the  top.  From  H 
draw  a  horizontal  line  and  complete  the  cube. 

The  extremes  of  foreshortening  are  shown  in 
Fig.  8. 

Copy  Figs.  7  and  8,  and  draw  the  cube  in  other 
positions  on  the  diagrams. 


CHAPTER  IL 


In  a  perspective  diagram  there  are  two  lines  which 
invariably  bear  to  each  other  the  same  geometric  rela¬ 
tion.  They  are  always  present  and  the  other  lines  and 
the  points  are  dependent  upon  them.  These  are  the 
Horizontal  Line  and  the  Principal  Yisnal  Ray.  The 
first  indicates  the  level  of  the  eye ;  the  second  is  the 
axis  of  tlie  cone  of  visual  rays,  and  extends  from  the 
eye  of  the  spectator  to  the  Horizontal  Line,  at  right 
angles  to  that  line.  (A  further  explanation  of  the 
cone  of  visual  rays  is  not  necessary  to  the  present  pur¬ 
pose.  It  may  be  found  under  Vision  in  any  cyclo¬ 
pedia.)  The  length  of  these  lines  is  arbitrary,  dej^end- 
ing  upon  convenience  or  special  necessity.  Wlien  the 
length  of  the  P.  Y.  R.  is  determined  upon,  however, 
it  decides  the  limits  of  the  drawing,  for  all  the  points 
on  the  diagram  are  determined  by  the  position  of  the 
Station  Point,  which  is  at  the  outer  end  of  this  line. 

Of  points  the  Station  Point,  which  represents  the 
position  of  the  eye  in  the  diagram,  is  fixed,  and  the 
Centre  of  Yision,  representing  the  point  opposite  the 
eye  in  the  horizon,  is  always  at  the  juncture  of  the 
P.  Y.  R.  with  the  Horizontal  Line. 

The  Yanishing  Point  is  any  point  in  the  diagram 
where  two  or  more  lines  meet  which  are  really  par¬ 
allel,  but  which  on  account  of  retreating  appear  to 
converge.  The  position  of  this  point  is  variable, 


18 


rEEa’ECTI  v^E. 


depending  on  the  direction  of  the  retreating  lines  of 
the  object  to  he  drawn. 

All  horizontal  retreating  lines  have  their  Vanishing 
Points  in  the  Horizontal  Line. 

The  Principal  Yisnal  Eaj,  although  drawn  verti¬ 
cally  in  order  to  represent  it  on  the  diagram,  represents 
a  horizontal  line  retreating  from  the  eye  at  an  angle 
of  90°.  Its  Vanishing  Point  is  in  the  Centre  of 
Vision.  As  all  parallel  retreating  lines  converge  to  a 
common  point,  it  follows  that  all  lines  parallel  to  the 


Principal  Visual  Kay:  i.  e.,  which  retreat  at  an  angle 
of  90°,  find  their  common  point  of  convergence  in  the 
Centre  of  Vision,  and  hence  the  Centre  of  Vision-is 
the  Vanishing  Point  for  all  such  lines. 

As  the  apparent  size  of  objects  varies  according  to 
distance  it  is  necessary  when  proportions  are  to  be 
measured  that  they  all  be  found  in  some  one  plane,  at 
a  chosen  distance  from  the  eye.  PerspecLve  assumes 
a  plane,  an  imaginary,  vertical  plane,  in  which  to 
measure  them.  It  is  called  the  Plane  of  Measuies.  A 


PEKSPECTIVE. 


19 


line  indicating  the  lower  edge  of  tins  plane  is  usually 
drawn  through  the  lower  line  or  angle  of  the  nearest 
object.  This  line  is  called  the  Ground  Line  and  is 
used  for  horizontal  measures. 

All  proportions  and  all  distances,  however  remote, 
can  be  measured  in  this  plane.  The  measurements  are 
called  Geometric  Scales,  because  they  give  the  actual 
proportions  of  the  parts  to  each  other.  The  only 
indication  of  this  plane  to  be  seen  in  the  diagram  is 
the  position  of  the  scales. 


There  is  still  another  imaginary  plane,  called  the 
Picture  Plane.  It  represents  the  surface  on  which 
the  drawing  is  supposed  to  be  made,  and  its  position 
marks  the  foreground  limits  of  the  picture.  The  Pict¬ 
ure  Plane  and  Plane  of  Measures  are  parallel  and  may 
coincide  or  not.  In  the  diagrams  here  given  they  are 
regarded,  for  simplicity,  as  one  and  the  same,  so  that 


20 


PERSPECTIVE. 


Plane  of  Measures  ”  and  “  Foreground  ’’  are  synon- 
omous  terms. 

The  Picture  Plane  is  indicated  at  Fig.  10,  where 
the  objects  rest  on  the  ground  with  their  nearest  sides 
against  it.  When  objects  touch  this  plane  they  are  in 
the  extreme  foregrounds  This  shows  the  Picture 
Plane  and  Plane  of  Measures  as  coincident,  the  lower 
line  of  the  Picture  Plane  being  used  as  the  ground 
Line  or  horizontal  scale  of  measures.  The  Plane  of 
Measures  may  be  assumed  at  any  distance  beyond  the 
Picture  Plane,  as  it  is  only  necessary  that  wherever  it 
is  all  the  scales  be  made  in  it,  in  order  to  be  propor¬ 
tional  ;  it  is  simpler,  however,  to  have  it  in  front  of 
the  objects,  and,  as  before  stated,  the  nearest  object  is 
usually  assumed  to  touch  it.  When  objects  do  not 
touch  the  Plane  of  Measures  their  scales  are  still 
drawn  in  it  and  transferred  to  the  distance,  as  will  be 
shown  in  the  next  lecture. 

Lines  parallel  to  the  Picture  Plane,  whatever  their 
direction — whether  vertical,  horizontal  or  oblique — re¬ 
main  parallel  to  it  when  put  into  perspective,  but  the 
length  of  such  lines  decreases  as  the  distance  increases. 
This  may  be  seen  by  practical  observation  and  by  re¬ 
ference  to  the  drawings  here  given. 

Lines  not  parallel  to  the  Picture  Plane  do  not  re¬ 
main  parallel  to  themselves,  but  appear,  instead,  to 
converge. 

When  an  object  is  so  placed  that  all  its  retreating 
lines  are  parallel  to  the  Principal  Yisual  Ray  the  ob¬ 
ject  is  said  to  be  in  Parallel  Perspective. 

The  Vanishing  Point  for  all  retreating  lines  in  Par¬ 
allel  Perspective  is  the  Centre  of  Vision. 


PEKSPECTIVE. 


21 


Lines  which  are  parallel  to  the  Picture  Plane  remain 
parallel  in  aj)pearance,  but  grow  shorter  as  they  retire 
into  the  distance.  It  is  easy  to  find  the  perspective 
of  such  lines  ;  as,  for  example,  take  the  nearest  line 
of  the  square  at  A,  Fig.  11.  If  lines  are  drawn  par¬ 
allel  to  it  at  any  distance  back  between  the  retreating 
lines  its  perspective  length  at  those  distances  will  be 
accurately  represented,  though  what  the  distances  are 
is  not  determined,  except  in  case  of  the  farther  line 
of  the  square,  which  is  definitely  placed. 

The  common  method  for  finding  perspective  dis¬ 
tances  was  described  in  the  last  lecture ;  following  is 
an  explanation  which,  if  thoroughly  understood,  will 
make  clear  the  whole  ground  of  elementary  perspec¬ 
tive. 

If,  as  we  have  seen,  a  line  drawn  from  the  Station 
Point  at  90°,  as  the  P.  Y.  R.,  in  Fig.  9  : — finds  at  its 
junction  with  the  Horizontal  Line  a  Vanishing  Point 
for  itself  and  all  lines  parallel  to  it,  so  a  line  drawn 
from  the  S.  P.  at  any  angle,  as  say  45°,  will,  in  cut¬ 
ting  the  Horizontal  Line,  determine  a  Vanishing  Point 
for  all  lines  running  in  that  direction.  fSTow  it  will  be 
found  that  horizontal  lines  directed  from  the  Station 
Point  at  an  angle  of  45°  will  strike  the  Measuring 
Points.  These,  therefore,  are  the  Vanishing  Points 
for  all  horizontal  lines  which  retreat  at  an  angle  of 
45°. 

If  the  sides  of  a  square  retreat  at  an  angle  of  90°, 
the  diagonals  of  the  square  will  retreat  at  an  angle  of 
45°,  therefore  the  Measuring  Points  are  the  Vanish¬ 
ing  Points  for  the  diagonals  of  the  square.  This  may 
be  tested  by  drawing  the  diagonals  of  the  horizontal 


22 


PERSPECTIVE. 


retreating  squares  which  have  been  obtained  by  the 
scale  as  described  in  the  last  lecture,  if  correctly  drawn 
their  diagonals  will  run  tov^ards  the  M.  P’s. 

In  Fig.  10  (A)  the  nearest  side  of  an  oblong  block 
being  drawn,  parallel  with  the  Picture  Plane  and 
touching  it,  draw  next  the  retreating  lines  to  the  Y. 
P.  It  is  now  necessary  to  find  the  perspective  wddth 
of  the  retreating  side.  Measure  out  from  the  nearest 
ground  angle  a  distance  equal  to  the  actual  width  of 
the  foreshortened  side.  This  measurement  is  the  Geo¬ 
metric  Scale.  From  the  end  of  this  scale  draw  a  line 
towards  the  M.  P.,  to  cross  the  adjacent  retreating  line 
of  the  object.  The  intersection  will  be  one  of  the 
farther  angles.  This  determines  the  distance  back 
from  the  Picture  Plane  of  the  two  remaining  lines  of 
the  object.  As  these  lines  do  not  retreat  draw  them 
parallel  to  the  corresponding  lines  in  the  foreground, 
between  the  retreating  lines  which  mark  their  lengths 
at  this  distance.  The  proportions  of  the  object  are  dif¬ 
ferent  but  there  is  no  difference  in  procedure  between 
the  drawing  of  these  and  the  cubes  at  Figs.  T  and  8. 

Suppose  the  point,  C,  to  remain  stationary  and  the 
block  to  be  turned  on  that  point  round  to  the  right, 
till  the  retreating  line,  D,  is  coincident  with  the  scale. 
It  will  be  readily  seen  that  the  angle  E  will  coincide 
with  the  end  of  the  scale,  G.  This  may  be  easily  de¬ 
monstrated  by  experiment. 

This  process  of  finding  perspective  distances  is  based 
on  a  simple  geometric  problem.  If  we  have  one  side 
of  an  isosceles  triangle  given,  and  wish  to  determine 
the  other  side  on  an  indefinite  line  we  may  do  so  by 
drawing  the  base  to  cut  the  indefinite  line  ;  the  inter- 


PERSPECTIVE. 


23 


section  will  determine  the  length  of  the  side  on  the 
indefinite  line.  The  Geometric  Scale  is  one  side  of 
an  isosceles  triangle,  the  retreating  line  from  C  to 
the  Vanishing  Point  is  another  side  whose  length  is 
not  determined,  the  line  from  the  end  of  the  scale 
towards  the  Measuring  Point  is  the  base  of  the  triangle, 
whose  direction  was  found  by  geometric  measurement 
at  the  Station  Point. 

The  lines  from  the  Vanishing  Point  to  the  Station 
Point  and  from  the  Vanishing  Point  to  the  Measur¬ 
ing  Point  are  two  legs  of  an  insosceles  triangle,  of 
which  a  line  from  the  Station  Point  to  the  Measuring 
Point  (shown  at  Fig.  10)  is  the  base,  and  are  parallel, 
respectively,  to  the  retreating  line  of  the  object,  the 
Geometric  Scale,  and  the  line  from  the  Geometric 
Scale  to  the  Measuring  Point.  For  the  geometric  re¬ 
lation  of  the  last  named  lines  see  (C)  Fig.  10. 

The  lines  from  the  S.  P.  to  the  M.  P’s  must  be 
regarded  as  being  in  the  same  horizontal  plane  as  the 
P.  V.  K.,  and  as  having  the  M.  P.  for  its  V.  P. 

In  parallel  perspective  the  base  of  the  triangle  is  at 
an  angle  of  45°  with  its  legs.  In  consequence  the 
diagonals  of  horizontal  squares  tend  towards  the 
Measuring  Points,  and  in  many  instances  it  is  more 
convenient  to  dispense  with  the  scale  and  find  the 
perspective  of  such  squares  by  drawing  their  diag¬ 
onals,  as  is  done  in  the  figures  which  follow. 

Fig.  11  (A)  is  a  square  plane  in  a  horizontal  position. 
The  farther  angles  are  determined  by  drawing  the 
diagonals.  In  this  case  the  diagonal  is  the  base,  and 
the  adjacent  sides  are  the  legs,  of  the  triangle.  Fig.  11 
(B)  is  a  cube  whose  perspective  has  been  determined 


24 


PEESPEOTIYE. 


by  drawing  all  its  retreating  lines — tliougli  in  an  opaque 
object  but  three  would  be  seen — and  the  diagonals  of 
the  base.  The  diagonals  of  the  upper  side  would  have 
obtained  the  same  result  and  obviated  the  necessity  of 
making  a  transparent  figure,  but  the  ground  surface 
was  used  because  it  afforded  a  larger  angle,  being 
farther  from  the  TI.  L.,  and  the  intersections  could  be 
more  easily  seen. 


S.P. 


It  should  be  explained  that  in  making  scales  the 
nearest  ground  corner  is  used  only  because  it  gives  the 
largest  angle  and  thereby  secures  the  greatest  accuracy. 
If  the  object  is  above  the  Horizontal  Line  the  nearest 
U23per  corner  is  the  best  jDlace  for  the  scale,  and  if  the 
ground  line  of  the  object  rests  on  the  Horizontal  Line 
a  scale  at  the  top  is  necessary.  Either  of  the  other 
corners  might  be  used,  with  the  other  Measuring 
Point,  but  they  give  a  slighter  angle  and  so  offer  more 
difficulty  in  determining  the  point  of  intersection,  and 


PERSPECTIVE. 


25 


ill  case  of  one  of  tliem  it  would  be  necessary  to  make 
a  transparent  drawing,  as  that  at  (B)  Fig.  11.  If  a 
scale  for  the  side  C,  of  this  figure,  were  to  be  drawn 
out  to  the  right  from  the  angle  D,  and  a  line  drawn 
from  it  to  M.  P.  1,  it  would  pass  through  the  angle 
E,  already  obtained  in  another  way ;  this  proves  both 
processes. 

Fig.  12  is  a  square  floor  with  square  tiles  whose 
edges  are  parallel  to  the  edges  of  the  floor.  Having 


S.P, 


drawn  a  ground  line  set  ofi  upon  it  the  nearest  edge 
of  the  floor,  which  is  one-half  on  each  side  of  the  P. 
Y.  P.  From  each  end  of  the  measured  edge  draw 
retreating  lines  to  the  Y.  P.  Draw  diagonals  and 
complete  the  square.  Divide  the  nearest  edge  into 
four  equal  parts  and  draw  lines  from  the  points  so 
found  to  the  Y.  P.  Thus  we  have  the  retreating  edges 
for  all  the  tiles.  The  diagonals  of  the  large  square 
are  diagonals  also  of  the  small  squares  they  cross. 


26 


PEKSPECTIVE. 


Draw  horizontal  lines  at  the  intersections  and  com¬ 
plete. 

Fig.  13  is  a  rectangular  pyramid.  Draw  the  square 
base  as  in  Fig.  12.  To  draw  the  inclined  sides  it  is 
necessary  first  to  find  their  upper  terminus,  the  apex 
of  the  pyramid.  The  apex  is  at  the  upper  end  of  the 
axis.  The  position  of  the  axis  is  already  determined 
by  the  intersection  of  the  diagonals,  which  occurs  at 
the  perspective  centre  of  the  base,  but  its  length  is 
still  to  be  found  because,  being  removed  from  the 


foreground  half  the  width  of  the  base,  it  appears 
shorter  than  it  really  is.  According  to  the  rule  laid 
down,  its  actual  height  must  be  measured  in.  the  Plane 
of  Measures,  in  which  the  nearest  edge  of  the  base  is 
drawn.  If  the  axis  should  be  moved  to  the  fore¬ 
ground,  keeping  it  parallel  to  the  retreating  edges  of 
the  base,  it  would  touch  the  nearest  edge,  A.  E.,  at  C. 
In  this  position  it  would  show  its  true  height  and  pro¬ 
portion  to  the  base.  Draw  therefore  a  line  at  C  the 


PEESPECTIVE. 


2T 


actual  height  it  is  desired  to  make  the  axis.  From  its 
upper  end  draw  a  line  to  the  Y.  P.,  because  the  axis, 
which  is  now  in  the  foreground,  retreats  at  an  angle 
of  90°,  and  stops  at  the  centre  of  the  base.  Raise  a 
line  from  the  centre  of  the  base  to  meet  the  retreating 
line.  Finish  by  drawing  the  inclined  lines  from  the 
base  angles  to  the  apex.  It  is  obvious  that  the  per¬ 
spective  height  of  the  axis  could  have  been  found  as 
well  by  erecting  the  scale  at  either  foreground  angle 
of  the  base,  and  drawing  the  retreating  line  to  an  M. 
P.  In  this  case  the  axis  would  have  been  moved  to 
its  position  along  a  plane  at  an  angle  of  45°,  which  is 
the  plane  of  the  diagonal. 

The  student  is  recommended  to  draw  the  diagrams, 
placing  the  objects  in  different  positions  from  those 
given,  until  he  is  sure  that  he  understands  the  princi¬ 
ples  involved.  Drawings  should  be  made  larger  than 
the  plates  here  given  ;  not  less  than  five  inches  for  the 
length  of  the  P.  Y.  R.  He  is  urged  also  to  prove  the 
statements  made  herein  by  actual  observation.  A 
great  German  savant  has  said  that  Nothing  which 
comes  through  the  eyes  into  the  head  ever  goes  out,” 
and  this  is  true  at  least  to  the  extent  that  things  are 
better  remembered  when  seen  than  when  merely  heard 
of.  It  is  comparatively  easy  to  comprehend  a  subject 
in  the  abstract,  but  abstract  knowledge  of  Perspective 
is  not  enough  for  the  artist.  He  must,  in  homely 
metaphor,  eat,  drink  and  sleep  with  it ;  it  must  inform 
his  pencil  as  magic  informed  the  sword  of  Orlando, 
for  he  can  never  make  a  pictorial  line  in  which  it  is 
not  involved. 


CHAPTEE  III. 


It  lias  been  shown  that  the  appearance  of  objects  in 
any  position  and  at  any  distance  can  be  accurately 
measured  and  drawn,  and  a  simj)le  method  has  been 
explained  for  accomplishing  this  result.  What  we 
have  learned  may  be  summed  up  as  follows. 

The  direction  of  retreating  lines  in  Perspective  is 
found  by  determining  their  real  or  geometric  direc¬ 
tion  at  the  Station  Point,  as  see  Figs.  9  and  10,  and  in 
case  of  horizontal  retreating  lines,  which  only  w^e  have 
discussed,  producing  these  lines  from  the  Station 
Point  to  cut  the  Horizontal  Line,  the  intersections  deter¬ 
mining  the  direction  of  all  lines  in  the  object  which 
are  parallel  to  these  lines. 

The  foreshortening  of  retreating  lines  is  deter¬ 
mined  by  drawing  their  real  length  in  the  Plane  of 
Measures  in  the  position  they  would  occupy  if  pro¬ 
jected  forwai’d  to  that  plane.  These  lines  are  called 
Geometric  Scales. 

It  \vas  shown  also  that  the  perspective  direction  of 
the  diagonals  of  horizontal  squares  in  Parallel  Per¬ 
spective  is  towards  the  Measuring  Points,  and  that,  in 
consequence,  the  diagonals  may  be  used  to  determine 
the  farther  angles,  instead  of  dravdng  a  scale. 

It  may  be  remarked,  thougli  obvious,  that  the  Yan- 
ishing  Points  have  reference  only  to  the  direction  of 
lines  and  planes,  and  not  to  their  position.  Objects 


V.P. 

WP.1 _ c.v, _ _  m.p.2 


PEESPECTIVE. 


31 


however  placed  have  the  same  Yaiiishing  points  when 
their  lines  and  planes  are  parallel. 

It  has  seemed  more  logicahto  explain  at  the  outset 
tlie  means  by  which  the  Yanishing  Point  of  any  line 
may  be  found  though  in  fact  it  is  not  necessary  in 
Parallel  Perspective  to  measure  the  angles  of  the  re- 
treating  lines  at  the  Station  Point,  because  their  Yan¬ 
ishing  Point  is  known  beforehand,  and  the  order  of 
drawing  the  lines  of  the  diagram  is,  practically,  as  was 
described  and  illustrated  in  Chapter  I. 

It  remains  now  to  make  a  little  further  application 
of  what  has  been  learned  of  parallel  perspective  and 
to  speak  of  some  alternative  means  of  obtaining  the 
same  results. 

Fig.  14  shows  objects  adjacent  to  the  plane  of 
Measures,  and  others  removed  from  it,  and  is  intended 
to  illustrate  the  fact  that  all  objects  in  the  picture 
must  be  measured  in  the  same  plane  in  order  to  be 
imoportional. 

A  and  B  are  the  same  size  as  C  and  D  but,  being 
distant,  appear  smaller.  To  draw  A  draw  first  the 
real  length  of  its  vertical  edge  in  the  plane  of  Meas¬ 
ures,  G  H.  This  line  is  a  vertical  scale.  As  the  ob¬ 
ject  is  parallel  to  the  P.  Y.  B.,  draw  the  lines  from  the 
scale  to  the  C.  Y.  which  is  their  Y.  P.  Mark  off  on 
the  Ground  Line  a  scale  equal  to  the  distance  of  the 
first  upright  of  the  object  from  the  foreground,  G  K. 
Draw  a  line  from  the  end  of  the  horizontal  scale,  K, 
to  the  M.  P.  and  the  intersection  with  the  retreating 
ground  line  at  M  will  be  the  perspective  distance  of 
the  nearest  edge  of  the  object,  which  is  an  upright 
line,  M  M,  between  the  two  vanishing  lines.  Deter- 


32 


PERSPECTIVE. 


mine  upon  the  width  of  the  object  and  add  a  corre¬ 
sponding  width,  K  O,  to  the  horizontal  scale.  Obtain 
the  farther  angle  in  the  same  manner  as  the  first  one. 
Draw  the  vertical  edges  between  the  retreating  lines 
and  complete  the  object.  Proceed  in  same  way  for 
the  horizontal  plane,  B.  If  it  is  desired  to  draw 
planes  still  more  distant,  add  on  to  the  horizontal  scale 
the  required  distance,  always  measuring  from  the 
point  in  the  plane  of  Measures  which  would  coincide 
with  the  point  sought  if  the  point  were  projected 
forward. 

There  is  another  method  of  finding  perspective  dis¬ 
tances  used  sometimes  in  place  of  the  one  we  have 
described.  It  is  called  the  method  of  Diagonals  and 
is  based  on  the  following  propositions : 

1.  A  line  drawn  through  the  intersection  of  the 
diagonals  of  a  parallelogram  bisects  the  parallelogram. 
This  is  the  common  way  of  dividing  a  perspective  sur¬ 
face  into  halves,  and  in  Fig.  16  (A)  determines  the 
centre  line  of  the  gables. 

2.  If  a  line  drawn  from  the  corner  of  a  parallelo¬ 
gram  to  the  middle  of  one  of  the  opposite  sides,  as  in 
(A),  Fig.  15,  and  continued  to  meet  the  other  side 
produced,  the  intersection  with  the  last  line  will  be  a 
farther  angle  for  another  equal  and  similar  parallelo¬ 
gram  adjacent  to  the  first. 

Suppose  an  object,  as  the  nearest  block  in  the  row 
in  Fig.  15,  A,  to  be  drawn  by  means  of  the  scale,  and 
it  is  desired  to  draw  another,  or  a  row  of  similar  blocks, 
adjacent.  Divide  the  nearest  upright  line  of  the  object 
in  the  middle,  B,  and  draw  a  line  from  the  point  thus 
made  to  the  Y.  P.  This  line  w/  1  bisect  the  side  of  the 


M'.P.I _  C.V.  H.L.  M.p.2 


PERSPECTIVE. 


35 


block,  and  of  every  block  which  may  be  drawn  beyond. 
Draw  a  line  from  one  of  the  nearest  angles,  C,  of  the 
block  to  cut  the  middle  of  the  opposite  side,  D,  and 
produce  to  cut  the  lower  retreating  line,  which  will 
give  E.  E  will  be  the  farther  ground  angle  of  the 
next  block.  This  is  a  shorter  method  than  drawing  a 
continuous  number  of  horizontal  scales.  But  if  the 
objects  are  not  adjacent  it  is  most  convenient  to  use 
the  scales.  The  square  planes  on  the  right  are  obtained 
by  drawing  their  diagonals  to  the  M.  P.,  but  they 
could  as  well  have  been  made  by  drawing  a  dividing 
line  to  the  Y.  P.,  and  proceeding  as  for  the  blocks  at 
(A).  If,  instead  of  squares,  it  were  desired  to  draw 
oblong  planes,  either  this  last  method  or  the  use  of 
the  scale  would  be  necessary,  unless  a  Y.  P.  were  found 
for  the  diagonals,  which  would  be  unnecessary  labor. 

In  Fig.  16  we  have  two  rows  of  gable-roofed  houses, 
showing  different  methods  of  obtaining  the  same  result. 

For  (A)  draw  first  the  nearest  side  of  the  nearest 
house,  A  B  C  D.  Draw  lines  from  A  and  B  to  the 
Y.  P.  Draw  a  scale,  B,  O,  for  the  width  of  the 
house  through  the  nearest  ground  corner.  By  means 
of  a  line  from  the  scale  to  the  M.  P.  locate  the  farther 
ground-angle  of  the  front,  S.  Draw  the  farther  up¬ 
right,  S.  T.  Draw  the  diagonals  of  the  front,  and 
from  their  intersection  raise  a  vertical  line  indefinite 
in  length.  The  apex  of  the  gable  will  be  found  in 
this  line  but  its  height  must  be  measured  by  a  verti¬ 
cal  scale  in  the  plane  of  measures,  as  in  case  of  the 
axis  of  the  pyramid  in  the  last  chapter.  A.  E.  is  such  a 
scale.  A  line  from  E  to  the  Y.  P.  will  give,  at  its  in¬ 
tersection  with  the  line  bisecting  the  gable,  the  height 


36 


PEKSPECTIVE. 


of  the  apex.  This  line,  from  the  vertical  scale  to  the 
y.  P.,  will  give  also  the  height  of  the  other  gables 
beyond.  Draw  the  inclined  lines,  and  complete  the 
front  of  the  house.  The  upper  line  of  the  roof,  or 
ridge-pole,  is  parallel  to  the  plane  of  measures,  but  as 
it  is  distant  from  it  half  the  width  of  the  house  it 
appears  shorter,  and  its  perspective  length  at  this  dis¬ 
tance  must  be  measured.  Erect  a  scale,  G,  from  the 
corner,  D,  and  draw  a  line  from  it  to  the  Y.  P.  Draw 
the  ridge-pole  to  cut  this  line  and  the  intersection 
will  determine  its  perspective  length.  From  the  point 
of  intersection,  H,  draw  the  inclined  line  to  D. 
Measure  off  horizontal  scales  on  the  ground  line  for 
the  widths  of  the  remaining  houses,  and  proceed  as  for 
the  first. 

For  (B),  draw  the  nearest  side  of  the  first  house, 
A  B  C  D.  Draw  lines  from  A  and  B  to  the  Y.  P. 
Place  the  further  upright  line  of  the  front  N.  P.  by 
means  of  a  scale,  B  E.  Draw  diagonals  of  the  front, 
and  at  their  intersection  draw  a  vertical  line  upwards, 
indefinite  in  length.  Make  a  vertical  scale  for  the 
height  of  the  roof,  AG.  A  line  from  this  scale  to 
the  Y.  P.  gives,  at  its  intersection  with  the  line  from 
the  diagonals,  the  height  of  the  apex.  Complete  the 
front  end  of  the  roof.  All  this  is  the  same  as  was 
done  at  (A).  The  slope  of  the  other  roof  lines  is  ob¬ 
tained  in  a  different  manner. 

If  a  sheet  of  paper  is  held  in  the  position  of  the 
slope  of  the  roof,  A  D  K  L,  it  will  be  seen  that  it  is 
not  parallel  to  the  picture  plane  but  inclines  upwards 
from  it,  and  that  as  the  upper  line  is  farther  away 
than  the  lower  one  it  appears  shorter  and  the  retreating 


.GR.L, 


i 


PERSPECTIVE. 


39 


ends  converge.  These  converging  ends,  A  K  and  D  L, 
must  have  a  Y.  P.,  and  if  it  is  found  we  will  have 
only  to  draw  the  slopes  towards  it  to  give  their  appar¬ 
ent  inclinations.  It  will  be  observed  that  these  lines 
lie  in  the  same  plane  as  the  back  and  front  of  the 
house,  which  retreats  at  an  angle  of  90°,  and,  therefore, 
if  they  were  horizontal  lines,  like  H,  their  Y.  P. 
would  be  in  the  C.  Y.,  but  they  are  not  horizontal 
but  incline  upwards,  instead,  and  so  their  Y.  P.  must 
be  found  by  looking  directly  ahead  and  raising  the 
eyes  above  the  C.  Y.  If  a  line  is  drawn  upwards 
from  the  C.  Y.,  the  Y.  P.  for  the  inclined  lines  will  be 
found  somewhere  in  it.  Where  depends  on  the  de¬ 
gree  of  inclination  of  the  roof,  which  must  be  deter¬ 
mined  beforehand.  In  this  case  the  inclination  of  the 
line,  A  K,  was  found  by  means  of  the  vertical  scale 
A  G,  and  the  diagonals  of  the  front.  If  this  line, 
A  K,  is  produced  to  cut  the  vertical  line  from  the 
C.  Y.  the  intersection  will  be  the  Y.  P.  sought,  and 
all  lines  parallel  to  A  K  must  be  drawn  towards  this 
Point.  Having  located  this  Y.  P.  draw  the  inclined 
line  from  D  and  finish  the  first  house.  The  width  of 
the  farther  houses  is  found  in  the  same  way  as  that  of 
the  blocks  in  Fig.  15.  The  line  from  the  vertical  scale, 
at  G,  to  the  Y.  P.  gives  the  height  of  each  roof,  and  its 
intersection  with  the  inclined  line  from  the  nearest 
corner  of  each  house  to  its  Y.  P.  determines  the  height 
of  the  roof  and  position  of  the  apex.  Finish  by  draw¬ 
ing  the  farther  slopes  of  the  roofs,  which  incline  equally 
in  the  opposite  direction.  A  Y.  P.  for  these  last 
slopes  would  be  found  below  the  level  of  the  eye  in 
the  P.  Y.  P.,  but  its  use  was  not  here  necessary. 


40 


PEESPECTIVE. 


Fig.  17  is  an  application  of  what  we  have  already 
learned  to  some  details  of  a  honse.  In  (A)  the  position 
of  the  windows  is  found  by  fixing  their  position  on  the 
scale  for  that  side  of  the  house  and  transferring  the 
points  to  the  retreating  ground  line  ;  their  height  was 
marked  on  the  near  end  of  the  house,  O,  and  a  line 
from  thence  to  the  Y.  P.  crossed  by  vertical  lines  from 
the  points  on  the  ground  line.  The  door  of  the  house 
(B)  was  found  in  the  same  way. 


To  draw  the  chimney  on  house  (A),  mark  its  width, 
height  and  position,  ABC,  on  the  end  of  the  near¬ 
est  gable,  which  is  in  the  Plane  of  Measures.  Draw 
lines  from  these  points  to  the  Y.  P.  Mark  its  dis¬ 
tance  from  the  Plane  of  Measures  and  the  width  of 
its  retreating  side  on  the  Geometrical  Scale  for  the 
retreating  side  of  the  house  at  E  F.  Transfer  these 


PERSPECTIVE. 


41 


points,  by  lines  to  tlie  M.  P.,  to  the  retreating  ground 
line.  Carry  lines  thence  to  the  roof  and  across  it, 
keeping  them  parallel  to  the  Plane  of  Measures. 
Where  these  lines  cut  the  line  from  B  to  the  Y.  P. 
we  have  the  position  of  two  lower  angles  of  the 
chimney.  Erect  two  vertical  lines  to  meet  the  retreat¬ 
ing  line  from  C.  This  completes  one  side  of  the 
chimney.  Draw  a  horizontal  line  from  G  to  cut  a  re¬ 
treating  line  from  A  at  H.  This  gives  the  width  of 
the  front,  H  G.  From  H  draw  a  vertical  line  to  meet 
a  horizontal  line  from  K.  The  oblique  line,  G  N, 
showing  the  insertion  of  the  chimney  into  the  roof 
is  parallel  to  the  other  oblique  lines  of  the  roof. 

Several  alternative  ways  will  suggest  themselves  for 
drawing  the  chimney  on  house  (B).  As  the  process 
shown  involves  only  what  has  been  already  explained, 
a  description  is  unnecessary. 

When  mechanical  accuracy  is  required  perspective 
work  is  reduced  from  actual  measurements  to  a  defi¬ 
nite  scale  of  inches  or  fractions  of  an  inch.  These  are 
set  off  in  the  Plane  of  Measures,  and  thus  the  smallest 
details  are  drawn  in  true  proportion.  Perspective 
problems  usually  state  the  scale  of  reduction.  Work¬ 
ing  to  a  scale  is  not  introduced  here  because  it  would 
cumber  the  student  while  being  no  help  towards  a 
knowledge  of  principles. 


CHAPTEE  lY. 


To  draw  any  curve  in  perspective  enclose  it  in  a  rec¬ 
tangle  ;  draw  tlie  perspective  of  the  rectangle ;  find 
the  points  of  contact  of  the  curve  with  the  rectangle ; 
draw  the  curve  through  these  points,  and  its  true  per¬ 
spective  will  be  found. 


If  the  curve  is  a  circle  the  rectangle  will  be  a  square, 
as  A  B  C  D,  Fig.  18,  and  the  points  of  contact  will 
be  the  ends  of  the  diameters,  E  F  Gr  H. 

If  it  is  desired  to  find  other  points  through  which  to 
draw  the  perspective  curve  not  in  the  rectangle,  find 
these  points  first  in  the  geometric  circle,  and  transfer 
them  to  the  perspective  circle  by  the  rule  already  given 
for  placing  a  point  in  any  given  perspective  position. 


PERSPECTIVE. 


4a 


viz  :  Fix  its  position  in  the  plane  of  measures,  as  A., 
Fig.  19,  and  draw  a  line  thence  towards  its  Y.  P. 
Cross  this  line  by  a  line  towards  the  M.  P.,  from  a 
scale  which  represents  the  actual  distance  of  the  point 
from  the  foreground  or  plane  of  measures,  and  the 
intersection  B  will  be  the  perspective  position  of  the 
point. 

Fig.  20  shows  the  circle  in  perspective.  A  descrip¬ 
tion  of  the  process  is  as  follows : 


Having  drawn  the  H.  L.,  P.  Y.  B.  and  G.  L.,  fix  the 
point  where  the  circle  is  to  come  in  contact  with  the 
Plane  of  Measures,  as  at  A.  Mark  off  on  the  G.  L.  the 
width  of  the  diameter  of  the  circle,  B  C,  one-half  on 
each  side  of  A,  and  draw  lines  from  thence  to  the  Y. 
P.  Draw  the  diagonals  and  complete  a  perspective 
square.  Draw  the  diameters.  The  ends  of  the  diame¬ 
ters,  A  F  G  H,  are  the  four  points  of  contact  of  the 
circle  with  the  square. 

It  is  desirable  for  greater  accuracy  to  find  points 
where  the  curve  crosses  the  diagonals.  These  are 


PEKSPECTIYE. 


U 

found  by  constructing  the  square,  or  half  of  it,  in  the 
plane  of  measures,  and  having  found  the  points  in 
this,  transferring  them  to  their  corresponding  places 
in  the  perspective  plan.  Construct  half  the  square 
with  B  C  as  one  side.  Draw  the  semi-diameters  and 
semi-diagonal.  Find  the  points  on  the  diagonals 
through  which  the  circle  passes  by  measuring  out  on 
them,  from  the  centre,  the  length  of  the  semi-diame¬ 
ters  or  by  inscribing  the  half  circle.  This  gives  points 


Fiy;.  so 

D  and  E.  Transfer  these  to  the  ground  line  at  I  and 
K  by  vertical  lines.  From  I  and  K  draw  lines  to  the 
Y.  P.,  and  where  these  cross  the  diagonals  of  the  per¬ 
spective  square  will  be  the  points  corresponding  to  D 
and  E.  Draw  the  curve  freehand  through  the  points 
obtained. 

If  it  is  desired  to  draw  on  the  diagram  a  circle 
removed  from  the  foreground,  construct  a  square  at 
the  desired  distance  and  proceed  as  for  the  first. 


PEKSPECTIVE. 


45 


A  circle  appears  as  such  only  when  the  eye  is  oppo¬ 
site  its  centre.  When  the  side  only  is  seen  it  is  a 
straight  line.  In  all  other  positions  it  is  an  ellipse. 
As  these  other  positions  are  practically  unlimited,  it 
oftenest  appears  as  an  ellipse. 

Fig.  21  shows  circles  in  an  upright  position  parallel 
to  the  plane  of  measures.  In  this  position  they  may 
be  drawn  without  the  aid  of  the  enclosing  rectangle, 
in  the  following  manner. 


v.p. 


Draw  the  nearest  circle  in  the  plane  of  measures, 
touching  the  G.  L.  Draw  line  A  B  from  the  cen¬ 
tre  to  the  point  of  contact  with  the  ground.  This 
is  the  radius  of  the  circle.  Draw  lines  from  A  and  B 
to  the  Y.  P.  A  will  pass  through  the  centre  of  all 
circles  which  may  be  drawn  beyond  and  B  will  pass 
through  their  point  of  contact  with  the  ground.  Make 
a  scale  from  B  the  actual  distance  of  the  farther  circle 
from  its  position  in  the  foreground.  A  line  from  the 
end  of  this  scale  to  the  M.  P.  will,  where  it  crosses 


46 


PEESPECTIVE. 


the  retreating  line  from  B,  at  O,  give  the  position  for 
the  farther  circle.  Draw  a  vertical  line  from  O  to  D. 
With  D  as  a  centre  and  the  distance  to  O  as  a  radius 
draw  the  circle. 

Circles  in  any  other  position  than  parallel  to  the 
picture  Plane  must  be  enclosed  in  squares.  Draw  the 
square  in  any  position  required  for  the  circle  and  then 
proceed  as  described  for  Fig.  20. 

Any  number  of  points  may  be  found  on  the  per¬ 
spective  circle  by  first  placing  them  on  the  geometric 
circle  in  the  plane  of  measures,  and  transferring  them, 
as  was  done  with  points  D  and  E. 

The  profile  lines  of  circular  solids,  as  cylinders, 
must  be  drawn  tangent  to  the  circles,  as  are  the  lines 
E  and  H  in  Fig.  21. 

The  forms  of  all  bodies  are  continuous  deviations 
from  geometric  forms  which  compose  their  bases,  and 
which,  when  imagined  round  them,  can  be  called  their 
general  outline.  To  determine  this  general  outline  is 
the  first  process  in  making  a  correct  drawing.  It  is 
found  by  circumscribing  the  object  as  closely  as  pos¬ 
sible  with  straight  lines  or  geometric  curves,  in  such  a 
way  that  we  complete  some  of  its  parts,  and  perhaps 
cut  off  some  of  its  ^protuberances.”  Such  an  outline 
will  represent  the  essential  form  and  character  of  the 
object. 

After  the  geometric  form  of  the  general  outline  is 
decided  upon,  it  must  be  drawn  in  the  perspective 
position  in  which  the  object  is  to  be  represented,  and 
afterwards  modified  into  the  object. 

Only  by  this  means,  by  reference  to  the  intentional 
structure,  can  disordered  forms  be  represented  truly  ; 


PEESPECTIVE. 


47 


for  if  the  normal  structure  is  recognized  the  departure 
from  it  will  seem  an  accidental  deviation,  but  if  the 
irregularities  are  drawn  without  this  guide  the  result 
will  have  neither  strength  nor  character.  The  one 
process  will  show  the  operation  of  mind  while  the 
other  will  be  chaos. 

The  perspective  of  flowers  which  are  based  on  the 
circle  will  serve  to  illustrate  this  point.  The  choice 
suggests  itself  from  the  fact  that  flowers,  while  being 
popular  objects  of  imitation  are  commonly  repre¬ 
sented  as  though  they  neither  occupied  solid  space  nor 
possessed  essential  form.  It  would  seem,  moreover, 
from  the  ordinary  portrayal,  that  they  were  exempt 
from  the  visual  laws  which  govern  the  appearance  of 
other  objects,  and  if  their  scientiflc  classiflcation 
were  to  be  determined  from  these  efflgies,  the  most 
expert  botanist  might  well  be  perplexed.  But  not¬ 
withstanding  that  ignorance  is  satisfied  by  variety 
without  order  the  flower  has  a  perfect  plan,  and  no 
one  can  hope  to  draw  it  correctly  by  scoring  down 
irregularities  as  essential  beauties,  without  reference  to 
the  law  which  is  behind. 

I  hold  this  rose  in  my  hand.  I  perceive  above  tlie 
straggling  leaves,  like  a  halo,  its  structural  plan,  the 
circle.  In  the  winged  life  of  each  petal  there  is  a  his¬ 
tory.  I  note  how  the  rain  beat  on  this  one  and  tlie 
sun  came  out  and  confirmed  its  downward  growth  ;  I 
see  the  trail  of  the  worm  on  that,  and  how  the  heat 
has  withered,  the  next,  and  the  countless  graceful  vol¬ 
untary  movements  of  the  parts  among  themselves. 
Scarcely  one  is  in  its  normal  place,  yet  it  is  by  refer¬ 
ence  to  the  normal  that  the  deviations  are  interesting. 


48 


PERSPECTIVE. 


The  irregularities  are  special  truths  and  mark  the  in¬ 
dividuality  of  each  leaf,  but  art  is  concerned  with 
types  more  than  with  special  truths,  and  the  artist  who 
rightly  depicts  a  flower  must  indicate  its  essential 
form,  and  must  show  that  the  variations  are  not  mere 
chance  without  organic  connection,  but  that  they  are 
temporary  escapings  from  a  law  to  which,  while  re¬ 
taining  their  individuality,  they  yet  submit.” 


Flowers  have  either  bi-lateral  symmetry,  like  the 
pansy,  or  the  stellar  symmetry  of  the  daisy.  The  plan 
or  general  outline  of  the  latter  class  is  the  circle.  The 
petals  radiate  from  the  centre  and  are  balanced  and 
equal.  Fig.  22  A  gives  the  construction  as  seen  from 
above  or  below,  the  radiating  lines  being  balancing 
lines  for  the  petals.  The  number  of  radiating  lines 


PEESPECTIVE. 


49 


differs  according  to  the  number  of  petals,  but  the  plan 
is  the  same  for  all  radiating  flowers. 

When  viewed  from  the  side  the  flower  is  balanced 
on  a  straight  line,  as  shown  at  B,  tho  line  at  the 
top,  G.  H.,  being  always  at  right  angles  to  the  balanc¬ 
ing  line,  E  C. 


These  two  positions,  A  and  B,  require  but  two  posi¬ 
tions  for  the  eye  with  relation  to  the  circle,  but  be¬ 
tween  these  two  there  are  an  inflnite  number  of 
appearances,  and  all  are  ellipses  of  varying  widths,  for 


50 


PERSPECTIVE. 


an  ellipse  may  be  any  width  between  a  straight  line 
and  a  circle,  these  two  being  the  perspective  poles  of 
the  curve. 


JTig.  S4: 

There  needs  yet  a  little  further  construction,  for  ij 
is  necessary  to  fix  not  only  the  outside  limit  for  the 
petals,  but  also  the  point  from  which  they  start,  E. 
When  the  outline  appears  a  circle  this  point  is,  of 
course,  the  center  of  the  circle,  as  shown  at  Fig.  22, 


and  when  the  view  is  a  side  one,  as  B,  Fig.  22,  the 
point  is  found  on  the  balancing  line  at  its  actual  dis¬ 
tance  from  the  outside  limit,  but  in  all  elliptical  views 
the  perspective  position  of  this  point  must  be  especi¬ 
ally  determined.  It  will  approach  the  centre  of  the 


PEKSPECTIVE. 


51 


curve  or  retire  from  it  according  as  the  ellipse  grows 
wider  or  narrower,  as  may  be  seen  at  Fig.  23.  The 
construction  is  shown  by  a  cone  at  Fig.  24.  Let  the 
base  of  the  cone  be  the  outline  of  the  flower  and  its 
centre  will  be  found  in  the  axis ;  not  necessarily  at 
the  apex  but  if  not  there  a  cross  section  must  be  drawn 
at  the  desired  distance  along  the  axis.  This  cross  sec¬ 


tion  will  be  another  ellipse  as  C  D,  and  will  enclose 
the  centre  or  heart  of  the  flower. 

To  draw  parallel  ellipses,  freehand,  draw  their 
diameters  parallel.  If  the  centre  of  one  ellipse  is  op¬ 
posite  the  centre  of  the  other,  as  occurs  in  the  flower, 
their  short  diameters  will  coincide.  Two  lines  con¬ 
necting  these  curves,  as  A  B,  B  C  at  Fig.  25,  will  be 
the  sides  or  proflles  of  the  flower.  If  these  sides  are 


52 


PERSPECTIVE. 


produced  to  meet,  it  will  be  seen  that  the  point  of 
meeting  is  opposite  the  centre  of  the  circles,  and  will 
fall  in  the  line  of  the  short  diameters  produced.  In 
the  flower  this  intersection  occurs  in  the  stem,  which 
is  always  a  prolongation  of  the  short  diameters. 

If  it  is  desired  to  draw  ellipses  a  deflnite  distance 
apart,  it  may  be  done  freehand  by  enclosing  them  in 
squares,  as  seen  at  Fig.  27,  the  distance  between  them 
being  measured  by  the  distance  between  the  nearest 
edges  of  the  squares,  as  from  A  to  B.  If  one  ellipse 


is  smaller,  its  size  and  position  are  set  off  in  the  nearest 
edge  of  the  square  in  whose  plane  it  is  to  be  drawn, 
as  C.  D.  Vanishing  lines  from  the  points  thus  made, 
crossing  the  diagonals  of  the  square  will  give  its  per¬ 
spective  position  and  size. 

Construction  lines  show  the  fundamental  form  of 
the  flower  or  other  object  which  they  circumscribe, 
and  its  perspective  position.  If  the  draughtsman  is 
expert  they  need  not  be  drawn  but  they  must  be  kept 
always  in  mind.  With  this  precaution  the  common 
mistakes  in  form  may  be  avoided. 


CHAPTEE  Y. 


We  have  thus  far  considered  objects  so  placed  that 
the  retreating  lines  are  at  an  angle  of  90°,  and  which 
being  thus  parallel  to  the  P.  Y.  K.  find  their  Y.  P.  in 
the  C.  Y. 

But  when  the  object  is  turned  to  left  or  right  so 
that  its  retreating  lines  are  no  longer  parallel  to  the  P. 
Y.  E.,  then  the  C.  Y.  is  no  longer  their  Y.  P.  Under 
this  change  the  object  is  said  to  be  in  angular  per¬ 
spective. 

As  the  retreating  lines  remain  horizontal  under  this 
change,  the  Y.  Ps  remain  in  the  Horizontal  Line,  but 
are  removed  to  a  distance  from  the  C.  Y.  depending 
upon  how  much  the  object  is  turned  from  the  plane  of 
the  picture. 

To  fix  the  Y.  P.  in  a  definite  place  it  is  necessary 
to  know  the  angle  at  which  the  lines  of  the  object  are 
turned  from  the  P.  P.  Distances  of  this  kind  are 
measured  by  degrees  of  a  circle.  For  example:  in  a 
quarter  circle  there  are  90°.  If  one  side  of  a  square 
is  parallel  with  the  line  C  D  (A),  Fig.  28,  then  the 
retreating  side  is  at  an  angle  of  90°  with  C  D.  This 
is  the  relative  position  of  the  object  to  the  picture 
plane  in  Parallel  Perspective.  But  if  it  is  turned  fur¬ 
ther  round  as  in  B,  the  angle  is  smaller  than  90° ; 
and  this  is  the  position  of  the  object  to  the  picture 
plane  in  angular  perspective.  How  much  smaller  it 


64: 


PERSPECTIVE. 


is  may  be  ascertained  by  measuring  the  number  of 
degrees  between  F  and  E. 

If  it  is  desired  to  measure  the  angle  at  which  the 
object  is  to  retreat  it  must  be  done  at  the  Station 
Pointj  as  explained  in  Chapter  II.  The  rule  is  as  fol¬ 
lows  :  Draw  a  line  from  the  8.  P.  at  any  angle  and 
produce  to  cut  the  II.  Z.,  and  the  intersection  will  he 
the  Y.  P.  for  all  lines  running,  in  that  direction. 


Therefore,  if  the  angle  at  which  the  lines  of  the  object 
retire  is  known  the  Y.  P.  may  be  fixed  as  follows. 

Suppose  the  angle  to  be  Y5°,  as  in  Fig.  29.  Having 
drawn  the  H.  L.  and  P.  Y.  H.,  draw  a  horizontal  line 
through  the  S,  P.  Draw  a  half  circle  on  this  last  line 
with  the  S.  P.  as  a  centre.  Measure  ofi  on  the  curve 
75°.  Draw  a  line  from  the  S.  P.  through  the  point 
thus  obtained,  C,  and  produce  to  cut  the  H.  L.,  and 
the  intersection  with  the  H.  L.  will  be  tlie  Y.  P. 


PERSPECTIVE. 


55 


sought.  Draw  the  nearest  upright  line  of  the  object, 
A  B,  resting  on  the  ground  line.  From  each  end 
draw  lines  to  the  Y.  P.  and  they  will  be  at  an  angle  of 
75°  with  the  Gr.  L.  or  Picture  Plane. 

When  the  angle  is  not  known,  the  direction  of  the 
retreating  line  may  be  determined  by  judgment  of 
the  eye,  thus :  Draw  the  nearest  upright  line  of  the 
object,  resting  on  the  Gr.  L.,  as  A.  B.  in  Fig.  30. 
Draw  a  retreating  line  from  A  to  meet  the  H.  L., 


H.L.  C.V.  \l.?. 


judging  its  direction  by  the  eye.  This  gives  the 
Y.  P.  for  this  line  and  all  others  parallel  to  it.  Draw 
a  line  from  B  to  the  same  Y.  P.  But  now,  in  com¬ 
pleting  the  cube,  it  will  be  seen  that  whereas  in 
parallel  perspective  all  the  retreating  lines  converged 
to  one  point,  here  they  do  not ;  for  the  lines  which 
formerly  were  parallel  to  the  Picture  Plane  now  re¬ 
treat,  and  in  a  different  direction  from  the  first  set. 
As  they  retreat  they  must  converge,  whereas  formerly 
they  did  not ;  as  they  converge  they  have  a  Y.  P.,  and 


56 


PEESPECTIVE. 


as  they  are  horizontal  their  Y.  P.  will  be  in  the  H.  L. 
The  Y.  P.  for  this  second  set  of  lines  is  as  far  from 
the  first  Y.  P.  as  the  actual  difierence  in  direction  of 
the  two  sets  of  lines,  the  angle  to  be  measured  at  the 
S.  P.  To  find  the  second  Y.  P.  draw  a  line  from  the 
first  Y.  P.  to  the  S.  P.,  and  make  with  it  there  an 
angle  equal  to  the  difierence  in  direction  of  the  two 
sets  of  lines.  In  case  of  the  cube,  the  lines  are  at 
right  angles,  therefore  make  a  right  angle  at  the  S.  P., 
and  produce  the  line  to  cut  the  H.  L.,  and  the  inter¬ 


section  will  be  the  Y.  P.  for  all  horizontal  lines  at 
right  angles  to  those  running  towards  the  first  Y.  P. 

Draw  retreating  lines  from  A  and  B  to  the  second 

Y.  P. 

It  is  necessary  now  to  find  the  M.  Ps.,  in  order  to 
measure  the  foreshortening  of  the  sides.  The  rule  for 
finding  the  M.  Ps.,  given  in  Chapter  I,  is :  Measure  the 
distance  from  the  Y,  P.  to  the  S.  P.^  and  marh  a 
jpoint  the  same  distance  from  the  P.  P.  on  the  II.L.; 
the  point  thus  found  is  the  M.  P. 


PERSPECTIVE. 


5r 

A  Y.  P.  needs  but  one  M.  P.  In  Parallel  Per¬ 
spective,  however,  the  Y.  P.  occupies  the  centre  of 
the  diagram,  and  it  is  convenient  to  have  the  M.  P. 
sometimes  on  one  side  and  sometimes  on  the  other, 
depending  on  the  position  of  the  object,  whether  to 
right  or  left  of  the  P.  Y.  K.  It  occupies  on  either 
side  the  same  relative  position,  being  used  to  right  or 
left  as  convenience  dictates.  Where  there  is  more 


S.P. 


than  one  Y.  P.  each  one  has  its  own  M.  P.,  used  ex¬ 
clusively  to  measure  the  foreshortening  of  the  lines 
which  vanish  towards  it.  The  M.  P.  for  any  Y.  P.  is 
found  when  an  isosceles  triangle  is  formed  with  the 
Y.  P.  as  the  apex,  the  line  to  the  S.  P.  as  one  -side, 
and  an  equal  side  laid  oil  on  the  H.  L. 

To  find  the  M.  Ps.  in  angular  perspective  take 
each  Y.  P.  as  a  centre  and  the  distance  to  the  S.  P.  as 


58 


PEESPECTIYE. 


a  radius,  and  strike  an  arc  to  cut  the  H.  L. ;  the  inter¬ 
sections  will  be  the  respective  M.  Ps. 

To  continue  the  drawing  of  Fig.  30,  measure  off  on 
the  Gr.  L.  a  scale  for  each  side  of  the  cube.  Draw  a 
line  from  each  to  the  M.  P.  for  that  set  of  lines  which 
it  is  to  measure.  At  the  intersections  with  the  lower 
retreating  lines  of  the  object  raise  vertical  lines  to 
meet  the  upper  retreating  lines.  This  completes  the 
vertical  sides.  It  now  remains  to  complete  the  top 


S.P. 


which  is  done  by  drawing  lines  from  the  upper  angles 
just  found,  one  to  each  Y.  P.,  the  intersection  of  these 
two  lines  being  the  farther  angle  of  the  top. 

It  will  be  seen  that  there  is  nothing  here  different  in 
procedure  from  parallel  perspective.  Another  Y.  P.  is 
added  because  the  Tines  which  formerly  were  drawn 
parallel  now  converge,  but  the  Y.  Ps.  are  used  in  pre¬ 
cisely  the  same  way,  and  the  M.  P.  for  each  is  found 


PERSPECTIVE. 


61 


by  precisely  the  same  rule  as  in  Parallel  Perspective. 
It  may  be  observed  that  in  Par.  Per.  the  isosceles  tri¬ 
angle  is  a  right  triangle,  because  the  Y.  P.,  which  is 
always  the  apex,  is  in  the  C.  Y.,  making  the  P.  Y.  P. 
one  of  its  sides,  but  when  the  Y.  P.  is  removed  from 
the  C.  Y.  the  P.  Y.  P.  is  no  longer  one  of  its  sides, 
and  the  angle  is  therefore  smaller. 

Fig.  31  shows  planes  in  angular  perspective.  But 
one  Y.  P.  is  needed.  The  first  three  are  measured  by 
scales,  and  the  others  by  the  method  of  Diagonals 
explained  in  Chapter  III. 

Fig.  32  shows  an  object  whose  sides  are  not  at  right 
angles  to  each  other,  and,  in  consequence,  whose  Y. 
Ps.  are  not  90°  apart. 

Draw  the  nearest  upright  line,  A  B.  Draw  one  of 
the  lower  retreating  lines  from  A  to  the  H.  L.,  deter¬ 
mining  its  direction  by  the  eye.  Draw  a  line  from 
the  Y.  P.  so  obtained  to  the  S.  P.  Make  an  angle  with 
this  line  equal  to  the  difference  in  direction  of  the 
sides  C  and  D  of  the  object.  In  this  case  the  sides 
are  60°  apart.  Produce  the  line  so  obtained  to  meet 
the  H.  L.  This  gives  the  Y.  P.  for  the  side  D.  Draw 
the  remaining  lines  from  A.  B.  to  the  Y.  P’s.  Find 
the  M.  P.  for  each  Y.  P.  Draw  the  scale  for  each 
side,  making  them  equal,  as  the  object  is  a  triangular 
prism.  By  lines  from  the  scales  to  the  respective  M. 
P’s.  find  the  foreshortening  of  each  side.  Paise  the 
vertical  lines  and  complete  the  sides.  As  there  are 
but  three  sides  to  the  object  finish  by  drawing  a  line 
connecting  the  upper  side  angles. 

The  method  of  finding  the  Y.  P.  for  inclined  lines 
was  explained  in  Chapter  III,  the  illustration  being  the 


62 


PEESPECTIVE. 


lines  of  a  roof.  The  Y.  P.  for  such  lines  is  found  in 
a  vertical  line  running  from  the  Y.  P.  of  the  plane  or 
side  in  which  the  lines  lie.  In  Pig.  16  they  were  in  a 
plane  which  had  its  Y.  P.  in  the  C.  Y.,  and  therefore 
their  Y.  P.  was  in  a  line  drawn  vertically  from  the  C. 
Y.  In  Fig.  33  the  inclined  lines  are  in  the  plane  which 
tends  towards  Y.  P.  1.  The  inclination  of  the  first 
line,  A,  may  be  assumed  or  it  may  be  measured  accu- 
jately  by  drawing  a  scale  for  the  height  of  the  roof, 
B,  and  drawing  a  line  thence  to  Y.  P.  1 ;  drawing  the 
diagonals  of  the  front  and  erecting  a  vertical  line  from 
their  centre  to  meet  the  line  from  the  scale  to  Y.  P.  1. 
Prolong  the  line  A  to  meet  the  vertical  line  from  Y. 
P.  1,  and  the  third  Y.  P.  will  be  found. 

When  an  object  is  so  placed  that  its  retreating  lines 
incline  up  or  down  from  the  horizontal,  as  does  the 
roof  of  Fig.  33,  the  object  is  said  to  be  in  oblique 
perspective. 

The  process  of  measuring  angles  as  at  the  S.  Ps.  in 
Figs.  29  and  32  is  geometrical  and  for  its  explanation 
the  student  is  referred  to  the  simple  problems  of  plane 
geometry. 


A  Practical  Book  on  PerspectiYO. 

Arclitectural  PersiifictiYe  for  Bepners. 

- - 

F.  A.  WRIGHT,  Architect. 

- - 

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COIVTONTS. 

Chap.  I— Introduction.  Chap.  II.— A  Small  Frame  House.  Chap. 
III.  A  Frame  Building.  Chap.  IV.— A  Brick  Building.  Chap.  V.— 
A  Stone  Building.  Chap.  'Sfl.—  The  Specifcations.  Chap.  VII.— 
Color.  By 

WILLIAM  B.  TUTHILL,  A.  M.,  ArcMtect. 

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I 


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